There are some results about solutions of Equation (REF ) on abelian groups in the literature. The case of \(G=\mathbb {R}\)  is treated by Kannappan [1]}. Stetkaer [2]} derives the solution formulas for the functional equation (REF ) on abelian groups. The method used in [3]}, [4]}, [5]}, [6]} proceeds from the starting point of the trigonometric functional equation
\(\mu (y) k(x \sigma (y))=k(x) l(y)-k(y) l(x), \quad x, y \in S.\) 
